Bielliptic curves and symmetric products
Bielliptic curves and symmetric products
复制标题
双椭圆曲线和对称积
DOI：
10.1090/s0002-9939-1991-1055774-0
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发表时间：
1991
期刊：
影响因子：
0.7
通讯作者：
J. Silverman
中科院分区：
文献类型：
--
作者：
J. Harris;J. Silverman
我们证明非超椭圆、非双椭圆曲线的双重对称积不包含任何椭圆曲线。应用法尔廷斯定理，我们得出这样的结论：在数域 K 上定义的曲线在 K 的所有二次扩展上只有有限多个点。我们用模曲线 Xo(N)、X1 (N)、X(N) 来说明我们的理论。如果一条平滑的投影曲线 C 允许 2 次映射 0: C -* X 到属零（相应为一）的曲线 X 上，则该曲线 C 称为超椭圆（相应为双椭圆）。在这种情况下，对称积
We show that the twofold symmetric product of a nonhyperelliptic, nonbielliptic curve does not contain any elliptic curves. Applying a theorem of Faltings, we conclude that such a curve defined over a number field K has only finitely many points over all quadratic extensions of K. We illustrate our theory with the modular curves Xo(N), X1 (N), X(N). A smooth, projective curve C is called hyperelliptic (respectively bielliptic) if it admits a map 0: C -* X of degree 2 onto a curve X of genus zero (respectively one). In this case the symmetric product